Given ^21C_15 ^10C_4=21!15!(21 15)! 10!4!(10 4)! #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160 ...

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Complex Numbers

Evaluate 21C...

Question

Evaluate
$^{21} C_{15} -^{10} C_{4}$

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Solution

Given
$^{21} C_{15} -^{10} C_{4} = \frac{21!}{15! (21 - 15)!} - \frac{10!}{4! (10 - 4)!}$

$= \frac{21!}{15! 6!} - \frac{10!}{4! 6!}$

$= \frac{21 \times 20 \times 19 \times 18 \times 17 \times 16 \times 15!}{15! 6!} - \frac{10 \times 9 \times 8 \times 6!}{4! 6!}$

$= \frac{21 \times 20 \times 19 \times 18 \times 17 \times 16}{6!} - \frac{10 \times 9 \times 8 \times 7}{4!}$

$= \frac{39070080}{720} - \frac{5040}{24}$

$= 54264 - 210$

$= 54054$

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